Lesson XIII Rationale of the Temperament. — (Concluded From Lesson XII.)
Proposition II.
That the student of scientific scale building may understand fully the reasons why the tempered scale is at constant variance with exact mathematical ratios, we continue this discussion through two more propositions, No. II, following, demonstrating the result of dividing the octave into four minor thirds, and Proposition III, demonstrating the result of twelve perfect fifths. The matter in Lesson XII, if properly mastered, has given a thorough insight into the principal features of the subject in question; so the following demonstration will be made as brief as possible, consistent with clearness.
Let us figure the result of dividing an octave into four minor thirds. The ratio of the length of string sounding a fundamental, to the length necessary to sound its minor third, is that of 6 to 5. In other words, 5/6 of any string sounds a tone which is an exact minor third above that of the whole string.
Now, suppose we select, as before, a string sounding middle C, as the fundamental tone. We now ascend by minor thirds until we reach the C, octave above middle C, which we call 3C, as follows:
Middle C-E♭; E♭-F♯; F♯-A; A-3C.
Demonstrate by figures as follows:Let the whole length of string sounding middle C be represented by unity or 1.
| E♭ will be sounded by 5/6 of the string | 5/6 |
| F♯, by 5/6 of the E♭ segment; that is, by 5/6 of 5/6 of the entire string, which equals | 25/36 |
| A, by 5/6 of 25/36 of entire string, which equals | 125/216 |
| 3C, by 5/6 of 125/216 of entire string, which equals | 625/1296 |
Now bear in mind, this last fraction, 625/1296, represents the segment of the entire string which should sound the tone 3C, an exact octave above middle C. Remember, our law demands an exact half of a string by which to sound its octave. How much does it vary? Divide the denominator (1296) by 2 and place the result over it for a numerator, and this gives 648/1296, which is an exact half. Notice the comparison.
| 3C obtained from a succession of exact minor thirds, | 625/1296 |
| 3C obtained from an exact half of the string | 648/1296 |
Now, the former fraction is smaller than the latter; hence, the segment of string which it represents will be shorter than the exact half, and will consequently yield a sharper tone. The denominators being the same, we have only to find the difference between the numerators to tell how much too short the former segment is. This proves the C obtained by the succession of minor thirds to be too short by 23/1296 of the length of the whole string.
If, therefore, all octaves are to remain perfect, it is evident that all minor thirds must be tuned flatter than perfect in the system of equal temperament.
The ratio, then, of 648 to 625 expresses the excess by which the true octave exceeds four exact minor thirds; consequently, each minor third must be flatter than perfect by one-fourth part of the difference between these fractions. By this means the dissonance is evenly distributed so that it is not noticeable in the various chords, in the major and minor keys, where this interval is almost invariably present. (We find no record of writers on the mathematics of sound giving a name to the above ratio expressing variance, as they have to others.)
Proposition III.
Proposition III deals with the perfect fifth, showing the result from a series of twelve perfect fifths employed within the space of an octave.
Method. Taking 1C as the fundamental, representing it by unity or 1, the G, fifth above, is sounded by a 2/3 segment of the string sounding C. The next fifth, G-D, takes us beyond the octave, and we find that the D will be sounded by 4/9 (2/3 of 2/3 equals 4/9) of the entire string, which fraction is less than half; so to keep within the bounds of the octave, we must double this segment and make it sound the tone D an octave lower, thus: 4/9 times 2 equals 8/9, the segment sounding the D within the octave.
We may shorten the operation as follows: Instead of multiplying 2/3 by 2/3, giving us 4/9, and then multiplying this answer by 2, let us double the fraction, 2/3, which equals 4/3, and use it as a multiplier when it becomes necessary to double the segment to keep within the octave.
We may proceed now with the twelve steps as follows:
Steps
| 1. | 1C | to | 1G | segment | 2/3 | for | 1G | |||||
| 2. | 1G | ″ | 1D | Multiply | 2/3 | by | 4/3, | gives | segment | 8/9 | ″ | 1D |
| 3. | 1D | ″ | 1A | ″ | 8/9 | ″ | 2/3 | ″ | ″ | 16/27 | ″ | 1A |
| 4. | 1A | ″ | 1E | ″ | 16/27 | ″ | 4/3 | ″ | ″ | 64/81 | ″ | 1E |
| 5. | 1E | ″ | 1B | ″ | 64/81 | ″ | 2/3 | ″ | ″ | 128/243 | ″ | 1B |
| 6. | 1B | ″ | 1F♯ | ″ | 128/243 | ″ | 4/3 | ″ | ″ | 512/729 | ″ | 1F♯ |
| 7. | 1F♯ | ″ | 1C♯ | ″ | 512/729 | ″ | 4/3 | ″ | ″ | 2048/2187 | ″ | 1C♯ |
| 8. | 1C♯ | ″ | 1G♯ | ″ | 2048/2187 | ″ | 2/3 | ″ | ″ | 4096/6561 | ″ | 1G♯ |
| 9. | 1G♯ | ″ | 1D♯ | ″ | 4096/6561 | ″ | 4/3 | ″ | ″ | 16384/19683 | ″ | 1D♯ |
| 10. | 1D♯ | ″ | 1A♯ | ″ | 16384/19683 | ″ | 2/3 | ″ | ″ | 32768/59049 | ″ | 1A♯ |
| 11. | 1A♯ | ″ | 1F | ″ | 32768/59049 | ″ | 4/3 | ″ | ″ | 131072/177147 | ″ | 1F |
| 12. | 1F | ″ | 2C | ″ | 131072/177147 | ″ | 2/3 | ″ | ″ | 262144/531441 | ″ | 2C |
Now, this last fraction should be equivalent to 1/2, when reduced to its lowest terms, if it is destined to produce a true octave; but, using this numerator, 262144, a half would be expressed by 262144/524288, the segment producing the true octave; so the fraction 262144/531441, which represents the segment for 2C, obtained by the circle of fifths, being evidently less than 1/2, this segment will yield a tone somewhat sharper than the true octave. The two denominators are taken in this case to show the ratio of the variance; so the octave obtained from the circle of fifths is sharper than the true octave in the ratio expressed by 531441 to 524288, which ratio is called the ditonic comma. This comma is equal to one-fifth of a half-step.
We are to conclude, then, that if octaves are to remain perfect, and we desire to establish an equal temperament, the above-named difference is best disposed of by dividing it into twelve equal parts and depressing each of the fifths one-twelfth part of the ditonic comma; thereby dispersing the dissonance so that it will allow perfect octaves, and yet, but slightly impair the consonance of the fifths.
We believe the foregoing propositions will demonstrate the facts stated therein, to the student’s satisfaction, and that he should now have a pretty thorough knowledge of the mathematics of the temperament. That the equal temperament is the only practical temperament, is confidently affirmed by Mr. W.S.B. Woolhouse, an eminent authority on musical mathematics, who says:
“It is very misleading to suppose that the necessity of temperament applies only to instruments which have fixed tones. Singers and performers on perfect instruments must all temper their intervals, or they could not keep in tune with each other, or even with themselves; and on arriving at the same notes by different routes, would be continually finding a want of agreement. The scale of equal temperament obviates all such inconveniences, and continues to be universally accepted with unqualified satisfaction by the most eminent vocalists; and equally so by the most renowned and accomplished performers on stringed instruments, although these instruments are capable of an indefinite variety of intonation. The high development of modern instrumental music would not have been possible, and could not have been acquired, without the manifold advantages of the tempered intonation by equal semitones, and it has, in consequence, long become the established basis of tuning.”
Numerical Comparison of the Diatonic Scale with the Tempered Scale.
The following table, comparing vibration numbers of the diatonic scale with those of the tempered, shows the difference in the two scales, existing between the thirds, fifths and other intervals.
Notice that the difference is but slight in the lowest octave used which is shown on the left; but taking the scale four octaves higher, shown on the right, the difference becomes more striking.
| Diatonic. | Tempered. | Diatonic. | Tempered. | |||
| C | 32. | 32. | C | 512. | 512. | |
| D | 36. | 35.92 | D | 576. | 574.70 | |
| E | 40. | 40.32 | E | 640. | 645.08 | |
| F | 42.66 | 42.71 | F | 682.66 | 683.44 | |
| G | 48. | 47.95 | G | 768. | 767.13 | |
| A | 53.33 | 53.82 | A | 853.33 | 861.08 | |
| B | 60. | 60.41 | B | 960. | 966.53 | |
| C | 64. | 64. | C | 1024. | 1024. | |
Following this paragraph we give a reference table in which the numbers are given for four consecutive octaves, calculated for the system of equal temperament. Each column represents an octave. The first two columns cover the tones of the two octaves used in setting the temperament by our system.
TABLE OF VIBRATIONS PER SECOND.
| C | 128. | 256. | 512. | 1024. |
| C♯ | 135.61 | 271.22 | 542.44 | 1084.89 |
| D | 143.68 | 287.35 | 574.70 | 1149.40 |
| D♯ | 152.22 | 304.44 | 608.87 | 1217.75 |
| E | 161.27 | 322.54 | 645.08 | 1290.16 |
| F | 170.86 | 341.72 | 683.44 | 1366.87 |
| F♯ | 181.02 | 362.04 | 724.08 | 1448.15 |
| G | 191.78 | 383.57 | 767.13 | 1534.27 |
| G♯ | 203.19 | 406.37 | 812.75 | 1625.50 |
| A | 215.27 | 430.54 | 861.08 | 1722.16 |
| A♯ | 228.07 | 456.14 | 912.28 | 1824.56 |
| B | 241.63 | 483.26 | 966.53 | 1933.06 |
| C | 256. | 512. | 1024. | 2048. |
Much interesting and valuable exercise may be derived from the investigation of this table by figuring out what certain intervals would be if exact, and then comparing them with the figures shown in this tempered scale. To do this, select two notes and ascertain what interval the higher forms to the lower; then, by the fraction in the table below corresponding to that interval, multiply the vibration number of the lower note.
Example. Say we select the first C, 128, and the G in the same column. We know this to be an interval of a perfect fifth. Referring to the table below, we find that the vibration of the fifth is 3/2 of, or 3/2 times, that of its fundamental; so we simply multiply this fraction by the vibration number of C, which is 128, and this gives 192 as the exact fifth. Now, on referring to the above table of equal temperament, we find this G quoted a little less (flatter), viz., 191.78. To find a fourth from any note, multiply its number by 4/3, a major third, by 5/4, and so on as per table below.
TABLE SHOWING RELATIVE VIBRATION OF INTERVALS BY IMPROPER FRACTIONS.
| The | relation | of the | Octave to a | Fundamental | is expressed by | 2/1 |
| ″ | ″ | ″ | Fifth to a | ″ | ″ | 3/2 |
| ″ | ″ | ″ | Fourth to a | ″ | ″ | 4/3 |
| ″ | ″ | ″ | Major Third to a | ″ | ″ | 5/4 |
| ″ | ″ | ″ | Minor Third to a | ″ | ″ | 6/5 |
| ″ | ″ | ″ | Major Second to a | ″ | ″ | 9/8 |
| ″ | ″ | ″ | Major Sixth to a | ″ | ″ | 5/3 |
| ″ | ″ | ″ | Minor Sixth to a | ″ | ″ | 8/5 |
| ″ | ″ | ″ | Major Seventh to a | ″ | ″ | 15/8 |
| ″ | ″ | ″ | Minor Second to a | ″ | ″ | 16/15 |
QUESTIONS ON LESSON XIII.
State what principle is demonstrated in Proposition II.
State what principle is demonstrated in Proposition III.
What would be the vibration per second of an exact (not tempered) fifth, from C-512?
Give the figures and the process used in finding the vibration number of the exact major third to C-256.
If we should tune the whole circle of twelve fifths exactly as detailed in Proposition III, how much too sharp would the last C be to the first C tuned?